Calculus Examples

$f (x) = x - \frac{1}{x^{2}}$ , $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

$y = (1) - \frac{1}{{(1)}^{2}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 1 - \frac{1}{1^{2}}$

Step 1.2.2

Remove parentheses.

$y = (1) - \frac{1}{{(1)}^{2}}$

Step 1.2.3

Simplify $(1) - \frac{1}{{(1)}^{2}}$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

One to any power is one.

$y = 1 - \frac{1}{1}$

Step 1.2.3.1.2

Cancel the common factor of $1$ .

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Step 1.2.3.1.2.1

Cancel the common factor.

$y = 1 - \frac{1}{1}$

Step 1.2.3.1.2.2

Rewrite the expression.

$y = 1 - 1 \cdot 1$

Step 1.2.3.1.3

Multiply $- 1$ by $1$ .

$y = 1 - 1$

Step 1.2.3.2

Subtract $1$ from $1$ .

$y = 0$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x - \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{x^{2}}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x^{2}}]$ .

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$1 - \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$1 - \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$1 - (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$1 - (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$1 - (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$1 - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.6

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 2.2.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.6.2

Multiply $2$ by $- 2$ .

$1 - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.7

Multiply $2$ by $- 1$ .

$1 - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.8

Raise $x$ to the power of $1$ .

$1 - (- 2 (x^{1} x^{- 4})) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.10

Subtract $4$ from $1$ .

$1 - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0$

Step 2.2.11

Multiply $- 2$ by $- 1$ .

$1 + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 2.2.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$1 + 2 x^{- 3} + 0$

Step 2.2.13

Add $2 x^{- 3}$ and $0$ .

$1 + 2 x^{- 3}$

Step 2.3

Simplify.

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Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + 2 \frac{1}{x^{3}}$

Step 2.3.2

Combine $2$ and $\frac{1}{x^{3}}$ .

$1 + \frac{2}{x^{3}}$

Step 2.3.3

Reorder terms.

$\frac{2}{x^{3}} + 1$

Step 2.4

Evaluate the derivative at $x = 1$ .

$\frac{2}{{(1)}^{3}} + 1$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

One to any power is one.

$\frac{2}{1} + 1$

Step 2.5.1.2

Divide $2$ by $1$ .

$2 + 1$

Step 2.5.2

Add $2$ and $1$ .

$3$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $3$ and a given point $(1, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 3 \cdot (x - (1))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y + 0 = 3 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Add $y$ and $0$ .

$y = 3 \cdot (x - 1)$

Step 3.3.2

Simplify $3 \cdot (x - 1)$ .

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Step 3.3.2.1

Apply the distributive property.

$y = 3 x + 3 \cdot - 1$

Step 3.3.2.2

Multiply $3$ by $- 1$ .

$y = 3 x - 3$

Step 4